The variation of invariant graphs in forced systems.

In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation is obtained depending on parameters, which involves the Markov map topological entropy. The results are illustrated with examples.

The mathematics of asymptotic stability in the Kuramoto model.

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.

Wave Generation in Unidirectional Chains of Idealized Neural Oscillators.

We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the far-downstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems. Here, we give a full mathematical proof of generation under uniform forcing in a simple piecewise affine setting for which the dynamics can be solved explicitly. In particular, our analysis proves existence, global stability, and robustness with respect to perturbations of the forcing, of families of waves with arbitrary period/wave number in some range, for every value of the parameters in the system.

Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering.

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.

Epigenetic aspects of lymphocyte antigen receptor gene rearrangement or 'when stochasticity completes randomness'.

To perform their specific functional role, B and T lymphocytes, cells of the adaptive immune system of jawed vertebrates, need to express one (and, preferably, only one) form of antigen receptor, i.e. the immunoglobulin or T-cell receptor (TCR), respectively. This end goal depends initially on a series of DNA cis-rearrangement events between randomly chosen units from separate clusters of V, D (at some immunoglobulin and TCR loci) and J gene segments, a biomolecular process collectively referred to as V(D)J recombination. V(D)J recombination takes place in immature T and B cells and relies on the so-called RAG nuclease, a site-specific DNA cleavage apparatus that corresponds to the lymphoid-specific moiety of the VDJ recombinase. At the genome level, this recombinase's mission presents substantial biochemical challenges. These relate to the huge distance between (some of) the gene segments that it eventually rearranges and the need to achieve cell-lineage-restricted and developmentally ordered routines with at times, mono-allelic versus bi-allelic discrimination. The entire process must be completed without any recombination errors, instigators of chromosome instability, translocation and, potentially, tumorigenesis. As expected, such a precisely choreographed and yet potentially risky process demands sophisticated controls; epigenetics demonstrates what is possible when calling upon its many facets. In this vignette, we will recall the evidence that almost from the start appeared to link the two topics, V(D)J recombination and epigenetics, before reviewing the latest advances in our knowledge of this joint venture.

Modeling temporal networks using random itineraries.

We propose a procedure to generate dynamical networks with bursty, possibly repetitive and correlated temporal behaviors. Regarding any weighted directed graph as being composed of the accumulation of paths between its nodes, our construction uses random walks of variable length to produce time-extended structures with adjustable features. The procedure is first described in a general framework. It is then illustrated in a case study inspired by a transportation system for which the resulting synthetic network is shown to accurately mimic the empirical phenomenology.

Clustering in cell cycle dynamics with general response/signaling feedback.

Motivated by experimental and theoretical work on autonomous oscillations in yeast, we analyze ordinary differential equations models of large populations of cells with cell-cycle dependent feedback. We assume a particular type of feedback that we call responsive/signaling (RS), but do not specify a functional form of the feedback. We study the dynamics and emergent behavior of solutions, particularly temporal clustering and stability of clustered solutions. We establish the existence of certain periodic clustered solutions as well as "uniform" solutions and add to the evidence that cell-cycle dependent feedback robustly leads to cell-cycle clustering. We highlight the fundamental differences in dynamics between systems with negative and positive feedback. For positive feedback systems the most important mechanism seems to be the stability of individual isolated clusters. On the other hand we find that in negative feedback systems, clusters must interact with each other to reinforce coherence. We conclude from various details of the mathematical analysis that negative feedback is most consistent with observations in yeast experiments.

Corepressive interaction and clustering of degrade-and-fire oscillators.

Strongly nonlinear degrade-and-fire (DF) oscillations may emerge in genetic circuits having a delayed negative feedback loop as their core element. Here we study the synchronization of DF oscillators coupled through a common repressor field. For weak coupling, initially distinct oscillators remain desynchronized. For stronger coupling, oscillators can be forced to wait in the repressed state until the global repressor field is sufficiently degraded, and then they fire simultaneously forming a synchronized cluster. Our analytical theory provides necessary and sufficient conditions for clustering and specifies the maximum number of clusters that can be formed in the asymptotic regime. We find that in the thermodynamic limit a phase transition occurs at a certain coupling strength from the weakly clustered regime with only microscopic clusters to a strongly clustered regime where at least one giant cluster has to be present.

TCR beta allelic exclusion in dynamical models of V(D)J recombination based on allele independence.

Allelic exclusion represents a major aspect of TCRbeta gene assembly by V(D)J recombination in developing T lymphocytes. Despite recent progress, its comprehension remains problematic when confronted with experimental data. Existing models fall short in terms of incorporating into a unique distribution all the cell subsets emerging from the TCRbeta assembly process. To revise this issue, we propose dynamical, continuous-time Markov chain-based modeling whereby essential steps in the biological procedure (D-J and V-DJ rearrangements and feedback inhibition) evolve independently on the two TCRbeta alleles in every single cell while displaying random modes of initiation and duration. By selecting parameters via fitting procedures, we demonstrate the capacity of the model to offer accurate fractions of all distinct TCRbeta genotypes observed in studies using developing and mature T cells from wild-type or mutant mice. Selected parameters in turn afford relative duration for each given step, hence updating TCRbeta recombination distinctive timings. Overall, our dynamical modeling integrating allele independence and noise in recombination and feedback-inhibition events illustrates how the combination of these ingredients alone may enforce allelic exclusion at the TCRbeta locus.

Spatial chaos of traveling waves has a given velocity.

We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy. These arguments provide a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.

Athermal dynamics of strongly coupled stochastic three-state oscillators.

We study the behavior of globally coupled ensembles of cyclic stochastic three-state units with transition rates from i-1 to i proportional to the number of units in state i. Contrary to mean-field theory predictions, numerical simulations show significant stochastic oscillations for sufficiently large coupling strength. The order parameter characterizing units synchrony increases monotonically with coupling while the coherence of oscillations has a maximum at a certain coupling strength. We find the exact formulas for the stationary probability distribution and the order parameter.

On the global orbits in a bistable CML.

In an infinite one-dimensional coupled map lattice (CML) for which the local map is piecewise affine and bistable, we study the global orbits using a spatiotemporal coding introduced in a previous work. The set of all the fixed points is first considered. It is shown that, under some restrictions on the parameters, the latter is a Cantor set, and we introduce an order to study the fixed points' existence. This also involves the proof of the coexistence of propagating fronts and stationary structures. In the second part, we analyze the global orbits which occur for strong coupling using the splitting of the dynamics into two independent (sub-)lattices, and emphasize the description of various traveling structures. (c) 1997 American Institute of Physics.

Kink dynamics in one-dimensional coupled map lattices.

We examine the problem of the dynamics of interfaces in a one-dimensional space-time discrete dynamical system. Two different regimes are studied: the non-propagating and the propagating one. In the first case, after proving the existence of such solutions, we show how they can be described using Taylor expansions. The second situation deals with the assumption of a travelling wave to follow the kink propagation. Then a comparison with the corresponding continuous model is proposed. We find that these methods are useful in simple dynamical situations but their application to complex dynamical behaviour is not yet understood. (c) 1995 American Institute of Physics.